Flat symplectic Lie algebras

Abstract

Let $(G,Ω)$ be a symplectic Lie group, i.e, a Lie group endowed with a left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call $(\G,ω=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined by $3ω\left(x\bullet y,z\right)=ω\left([x,y],z\right)+ω\left([x,z],y\right)$ extends to a left invariant connection $\na$ on $G$ which is torsion free and symplectic ($\na\Om=0)$. When $\na$ has vanishing curvature, we call $(G,Ω)$ a flat symplectic Lie group and $(\G,\om)$ a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to $\om$. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection $\na$ of a flat symplectic Lie group is always complete. We prove that the double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of double extension of flat symplectic Lie algebras starting from $\{0\}$. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension $\leq6$.